9  Permutations and Combinations

9.1 Introduction

This chapter focuses on counting techniques, essential for probability, arrangements, and selection problems.
We explore fundamental principles of counting, ordered arrangements (permutations), unordered selections (combinations), circular arrangements, restrictions, and practical exam applications.

9.2 1) Fundamental Counting Principles

  • Addition Rule: If event A can occur in \(m\) ways, and event B in \(n\) ways (mutually exclusive), total = \(m+n\).
  • Multiplication Rule: If event A can occur in \(m\) ways and B in \(n\) ways (independent, sequential), total = \(m \times n\).

Example: 3 shirts, 4 trousers → 3×4=12 outfits.

9.3 2) Permutations (Arrangements with Order)

9.3.1 2.1 Definition

Permutation = arrangement of objects in a specific order.

\[ nP_r = \frac{n!}{(n-r)!} \]

  • \(n! = n\times(n-1)\times\cdots\times1\)
  • By convention \(0! = 1\)

9.3.2 2.2 Special cases

  • Arrange \(n\) distinct objects: \(n!\) ways.
  • Arrange \(n\) objects, \(r\) at a time: \(nP_r\).
  • Arrange with repetition allowed: \(n^r\).

Example: Arrange 4 books on shelf: \(4!=24\) ways.

9.4 3) Combinations (Selections without Order)

9.4.1 3.1 Definition

Combination = selection where order does not matter.

\[ nC_r = \frac{n!}{r!(n-r)!} \]

9.4.2 3.2 Properties

  • \(nC_r = nC_{n-r}\)
  • \(nC_0=nC_n=1\)

Example: Choose 3 from 10 students: \(10C_3=120\).

9.5 4) Relation between \(nP_r\) and \(nC_r\)

\[ nP_r = nC_r \times r! \]

First choose r objects (\(nC_r\)), then arrange them (\(r!\)).

9.6 5) Circular Permutations

  • \(n\) objects in a circle: \((n-1)!\) arrangements.
  • If clockwise and anticlockwise considered same: \(\frac{(n-1)!}{2}\).

Example: Arrange 6 people around round table: \(5!=120\).

9.7 6) Permutations with Repetition (Non-distinct objects)

If \(n\) objects with groups of identicals:

\[ \frac{n!}{p_1!p_2!\dots p_k!} \]

Example: Word “BALLOON” = 7 letters with L twice, O twice → \(\frac{7!}{2!2!}=1260\).

9.8 7) Restricted Arrangements and Selections

9.8.1 7.1 Together/Separated

  • If A and B must be together: treat them as one unit.
  • If A and B must not be together: total − (together cases).

9.8.2 7.2 Distribution of objects

  • Identical balls into distinct boxes: use stars and bars (advanced).
  • Distinct balls into distinct boxes: each has choices, multiply.

9.9 8) Practical Applications

  1. Form 4-digit numbers from digits without repetition.
  2. Select committees from groups.
  3. Word formation from letters.
  4. Arrangements with restrictions (vowels together, certain persons not adjacent).
  5. Exam DI puzzles.

9.10 9) Solved Examples

Ex 1: Number of 3-digit numbers from digits 1–5 without repetition.
= \(5P3=60\).

Ex 2: Choose a president, VP, secretary from 10 people.
= \(10P3=720\).

Ex 3: Committee of 3 from 8 persons.
= \(8C3=56\).

Ex 4: Arrange letters of “MATH” with vowels together.
Treat (A) as one unit → 3! ways ×2! (arrange vowels internally) = 12.

Ex 5: Circular arrangement of 7 persons, clockwise ≠ anticlockwise.
= (7−1)! = 6! = 720.

Ex 6: Word “MISSISSIPPI”.
Total = \(\frac{11!}{4!4!2!}=34,650\).

9.11 10) Common Traps

  • Confusing order vs selection.
  • Forgetting to adjust for identicals.
  • Miscounting circular permutations.
  • Not distinguishing “at least one” vs “exactly one”.
  • Overcounting when restrictions apply.

9.12 Practice Set – Level 1

  1. Form 3-digit numbers from 1–9 without repetition.
  2. Arrange 5 students in a line.
  3. Choose 2 from 6 persons.
  4. Arrange 7 persons in circle.
  5. Number of ways to arrange “LEVEL”.

9.13 Practice Set – Level 2

  1. Word “BANANA” arrangements.
  2. Committee of 4 from 10 students, one must be a girl (out of 4 girls, 6 boys).
  3. 4-digit numbers from 1–9 with repetition allowed.
  4. Distribute 4 identical balls into 3 boxes.
  5. Seating 3 boys, 3 girls alternately in row.

9.14 Practice Set – Level 3 (Challengers)

  1. Number of ways to seat 5 couples around table, no spouses adjacent.
  2. Word “PROBABILITY”: vowels together.
  3. Select team of 5 from 8 men + 5 women, at least 2 women.
  4. Permutations of digits 0–9 to form 4-digit even numbers.
  5. 6-digit numbers from digits 1–6, divisible by 6.

9.15 Answer Key (Outline)

Level 1: 1) 504; 2) 120; 3) 15; 4) 720; 5) 30.
Level 2: 6) 60; 7) 210; 8) 9⁴=6561; 9) 15; 10) 36.
Level 3: 11) 1152; 12) 1814400; 13) 2002; 14) 4536; 15) 120.

9.16 Summary

  • Use nPr for arrangements, nCr for selections.
  • Relation: \(nPr=nCr\times r!\).
  • Circular = \((n−1)!\) (or half if reflection same).
  • Identicals → divide by factorials.
  • Carefully apply restrictions.
    With practice, problems collapse into applying right formula quickly.